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A quadratic space is a real vector space V with a quadratic form Q(x), e.g. V = R^n with Q as the squared length. The Clifford algebra Cl(V) of a quadratic space is the associative algebra that contains V and satisfies x^2 = Q(x) for all x in V. We're imposing by fiat that the square of a vector should be the quadratic form's value and seeing where it takes us. Treat x^2 = Q(x) as a symbolic rewriting rule that lets you replace x^2 or x x with Q(x) and vice versa whenever x is a vector. Beyond that Cl(V) satisfies the standard axioms of an algebra: it lets you multiply by scalars, it's associative and distributive, but not necessarily commutative.

Remarkably, this is all you need to derive everything about Clifford algebras.

Let me show you how easy it is to bootstrap the theory from nothing.

We know Cl(V) contains a copy of V. Since x^2 = Q(x) for all x, it must also contain a copy of some nonnegative reals.

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protected async void Page_Load(object sender, EventArgs e)
{
    test++;
    if (!Page.IsPostBack)
    {
        if (Request.Form["method"] == "Test")
        {
            MyType[] obj = await Test(Request.Form["msg1"], Request.Form["msg2"]);
            string json = new JavaScriptSerializer().Serialize(obj);
            Response.Write(json);

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open System

type Eff<'Ctx, 'T> = 'Ctx -> 'T

type EffBuilder() =
	member __.Return x : Eff<'Ctx,'T> = 
		fun _ -> x 
	member __.Bind(f : Eff<'Ctx, 'T>, g : 'T -> Eff<'Ctx, 'S>) : Eff<'Ctx, 'S> = 
		fun c -> g (f c) c 
	member __.Zero() : Eff<'Ctx, unit> = 

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let rec bfs2robust (fanout: Map<'node, 'node seq> -> 'node -> 'node seq) (tree: Map<'node, 'node seq>) (node: 'node) : 'node seq = 
    let single = seq [node]
    match fanout tree node with
    | e when e = Seq.empty -> single
    | s -> Seq.fold (fun acc item -> 
        bfs2robust fanout (tree |> Map.remove node) item 
        |> Seq.append acc) single s |> Seq.distinct

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let rec bfs2 (fanout: Map<'node, 'node seq> -> 'node -> 'node seq) (tree: Map<'node, 'node seq>) (node: 'node) : 'node seq = 
    let single = seq [node]
    match fanout tree node with
    | e when e = Seq.empty -> single
    | s -> Seq.fold (fun acc item -> 
        bfs2 fanout tree item 
        |> Seq.append acc) single s

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public class BindableTextBlock : TextBlock
{
    public ObservableCollection<Inline> InlineList
    {
        get { return (ObservableCollection<Inline>)GetValue(InlineListProperty); }
        set { SetValue(InlineListProperty, value); }
    }
    
    public static readonly DependencyProperty InlineListProperty =
        DependencyProperty.Register("InlineList", typeof(ObservableCollection<Inline>), typeof(BindableTextBlock), new UIPropertyMetadata(null, OnPropertyChanged));

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            <DataGrid Grid.Column="2" ItemsSource="{Binding SelPayment.eur_invoices}" IsReadOnly="True" AutoGenerateColumns="False"
                HorizontalAlignment="Stretch" VerticalAlignment="Stretch" Margin="10,10,10,10" >
                <DataGrid.Columns>
                    <DataGridTextColumn Header="Buy/Sell" Binding="{Binding buysell}" />
                    <DataGridTextColumn Header="Cpty Name" Binding="{Binding CptyName}" />
                    <DataGridTextColumn Header="Invoice Nr" Binding="{Binding InvoiceNr}" />
                    <DataGridTextColumn Header="Amount" Binding="{Binding amount, StringFormat=N}" />
                    <DataGridTextColumn Header="Curr" Binding="{Binding curr}" />
                    <DataGridTextColumn Header="Msg" Binding="{Binding ui.msg}" />
                </DataGrid.Columns>

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type Back2UI() = 
    inherit SimpleViewModelBase() 
    let mutable _msg = ""
    member this.msg
        with get() = _msg
        and set value =
            _msg <- value
            this.OnPropertyChanged(<@ this.msg @>)

type ActivePassive = Active | Passive

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<DataGrid Grid.Column="2" ItemsSource="{Binding SelPayment.eur_invoices}" IsReadOnly="True"
                HorizontalAlignment="Stretch" VerticalAlignment="Stretch" Margin="10,10,10,10" >

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// business model
type Invoice = { CptyName: string; CptyNum: int; InvoiceNr: string; amount : decimal; curr: string; buysell: ActivePassive}

// array extraction from Excel with type provider
let ActiveInvoice = 
    ActiveInvoiceExcel.Data
    |> Seq.map ( fun line -> { InvoiceNr = line.``Numero documento``; ... })
    |> Seq.toArray